Problem
NT-B2-M12-P002 Two Power Divisibilities
#2
★★☆☆☆ Level 2 of 5
Prove that \(a-b\mid a^n-b^n\) and \(a+b\mid a^{2n+1}+b^{2n+1}\) for all integers \(a,b\) and \(n\ge0\).
In the second part, replace \(b\) by \(-b\).
The first statement follows from the difference of powers formula. For the second, \(a^{2n+1}+b^{2n+1}=a^{2n+1}-(-b)^{2n+1}\), which is divisible by \(a-(-b)=a+b\).
This works well as a short warm-up before more general polynomial problems.